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Characterizing Finite Frobenius RingsVia Coding Theory
Jay A. Wood
Department of MathematicsWestern Michigan University
http://homepages.wmich.edu/∼jwood/
Algebra and Communications SeminarUniversity College Dublin
November 7, 2011
Florence Jessie MacWilliams
I 1917–1990
I Bell Labs
I 1962 Harvard dissertation under Andrew Gleason:“Combinatorial Problems of Elementary AbelianGroups”
I Three sections:I Extension theorem on isometriesI The MacWilliams identitiesI Coverings
JW (WMU) Finite Frobenius Rings November 7, 2011 2 / 40
Linear Codes Defined over Finite Rings
I Let R be a finite ring with 1. A linear code of lengthn defined over R is a left R-submodule C ⊂ Rn.
I There were some results on codes over rings in the1970s, but the real breakthrough came in 1994.Hammons, Kumar, Calderbank, Sloane, and Soleshowed that important duality properties of certainnon-linear binary codes could be explained by linearcodes defined over Z/4Z.
I Are the fundamental results of MacWilliams validover finite rings?
JW (WMU) Finite Frobenius Rings November 7, 2011 3 / 40
Code Equivalence
I When should two linear codes be considered thesame?
I Monomial equivalence (external)
I Linear isometries (internal)
I These notions are the same over finite fields: theMacWilliams extension theorem.
JW (WMU) Finite Frobenius Rings November 7, 2011 4 / 40
Monomial equivalence
I Work over a finite ring R .
I A permutation σ of {1, . . . , n} and invertibleelements (units) u1, . . . , un in R determine amonomial transformation T : Rn → Rn by
T (x1, . . . , xn) = (xσ(1)u1, . . . , xσ(n)un).
I Two linear codes C1,C2 ⊂ Rn are monomiallyequivalent if there exists a monomial transformationT such that C2 = T (C1).
JW (WMU) Finite Frobenius Rings November 7, 2011 5 / 40
Linear Isometries
I The Hamming weight wt(x) of a vectorx = (x1, . . . , xn) ∈ Rn is the number of nonzeroentries in x .
I A linear isomorphism f : C1 → C2 between linearcodes C1,C2 ⊂ Rn is an isometry if it preservesHamming weight: wt(f (x)) = wt(x), for all x ∈ C1.
I If T is a monomial transformation with C2 = T (C1),then the restriction of T to C1 is an isometry.
I Is the converse true? Does every linear isometrycome from a monomial transformation?
JW (WMU) Finite Frobenius Rings November 7, 2011 6 / 40
MacWilliams Extension Theorem overFinite Fields
Assume C1,C2 are linear codes in Fnq. If a linear
isomorphism f : C1 → C2 preserves Hamming weight,then f extends to a monomial transformation of Fn
q.
I MacWilliams (1961); Bogart, Goldberg, Gordon(1978)
I Ward, Wood (1996)
JW (WMU) Finite Frobenius Rings November 7, 2011 7 / 40
Generalizing the Work of MacWilliams
I When A = R , is the MacWilliams extension theoremstill valid?
I Yes, if R is a finite Frobenius ring.
I Why Frobenius?
I There is a character-theoretic proof over finite fieldsthat uses the crucial property F ∼= F.
I Frobenius rings satisfy R ∼= R , and the same proofwill work.
JW (WMU) Finite Frobenius Rings November 7, 2011 8 / 40
Characters of Finite Abelian Groups
I Let (G ,+) be a finite abelian group.
I A character π of G is a group homomorphismπ : (G ,+)→ (C×,×), where (C×,×) is themultiplicative group of nonzero complex numbers.
I Example: let G = Z/nZ be the integers modulo n.For any a ∈ Z/nZ, πa(x) = exp(2πiax/n), x ∈ G , isa character of G .
I Example: let G = Fq. For any a ∈ Fq,πa(x) = exp(2πi Tr(ax)/p), x ∈ Fq, is a characterof Fq. (Tr : Fq → Fp is the absolute trace to theprime subfield.)
JW (WMU) Finite Frobenius Rings November 7, 2011 9 / 40
Character Groups
I The set G of all characters of G is itself a finiteabelian group called the character group.
I |G | = |G |.I As elements of the vector space of all functions from
G to C, the characters are linearly independent.
I If M is a finite left R-module, then M is a rightR-module.
JW (WMU) Finite Frobenius Rings November 7, 2011 10 / 40
Two Useful Formulas
∑x∈G
π(x) =
{|G |, π = 1,
0, π 6= 1.
∑π∈G
π(x) =
{|G |, x = 0,
0, x 6= 0.
JW (WMU) Finite Frobenius Rings November 7, 2011 11 / 40
Finite Frobenius Rings
I Finite ring R with 1.
I The (Jacobson) radical Rad(R) of R is theintersection of all maximal left ideals of R ; Rad(R)is a two-sided ideal of R .
I The (left/right) socle Soc(R) of R is the ideal of Rgenerated by all the simple left/right ideals of R .
I R is Frobenius if R/Rad(R) ∼= Soc(R) as one-sidedmodules (both left and right).
JW (WMU) Finite Frobenius Rings November 7, 2011 12 / 40
Two Useful Theorems About FiniteFrobenius Rings
I (Honold, 2001) R/Rad(R) ∼= Soc(RR) as leftmodules iff R/Rad(R) ∼= Soc(RR) as right modules.
I R is Frobenius iff R ∼= R as left modules iff R ∼= Ras right modules (Hirano, 1997; indep. 1999).
I Corollary: R is Frobenius iff there exists a characterπ of R such that ker π contains no nonzero left(right) ideal of R . This π is a generating character.
JW (WMU) Finite Frobenius Rings November 7, 2011 13 / 40
Examples of Finite Frobenius Rings
I Finite fields Fq: π(x) = exp(2πi Tr(x)/p).
I Z/nZ: π(x) = exp(2πix/n).
I Galois rings (Galois extensions of Z/pmZ).
I Finite chain rings (all ideals form a chain).
I Products of Frobenius rings.
I Matrix rings over a Frobenius ring: Mn(R).
I Finite group rings over a Frobenius ring: R[G ].
I F2[X ,Y ]/(X 2,XY ,Y 2) is not Frobenius (Klemm,1989).
JW (WMU) Finite Frobenius Rings November 7, 2011 14 / 40
MacWilliams Extension Theorem overFinite Frobenius Rings
Theorem (1999)Let R be a finite Frobenius ring, and supposeC1,C2 ⊂ Rn are left linear codes. If f : C1 → C2 is anR-linear isomorphism that preserves Hamming weight,then f extends to a monomial transformation of Rn.
I Also, Greferath and Schmidt (2000), using posettechniques.
I Greferath (2002), generalizing Bogart, et al.
JW (WMU) Finite Frobenius Rings November 7, 2011 15 / 40
Character-Theoretic Proof (a)
I The proof follows a proof of Ward and Wood in thefinite field case (1996).
I View Ci as the image of Λi : M → Rn, withΛi = (λi ,1, . . . , λi ,n) and Λ2 = f ◦ Λ1.
I Using character sums, express Hamming weight as:
wt(Λi(x)) = n −n∑
j=1
1
|R |∑π∈R
π(λi ,j(x)), x ∈ M .
JW (WMU) Finite Frobenius Rings November 7, 2011 16 / 40
Character-Theoretic Proof (b)
I Because f preserves Hamming weight, we get
n∑j=1
∑π∈R
π(λ1,j(x)) =n∑
k=1
∑ψ∈R
ψ(λ2,k(x)), x ∈ M .
I In a Frobenius ring, there is a generating characterρ. Every character of R has the form aρ, a ∈ R .
I (aρ)(r) := ρ(ra), r ∈ R .
JW (WMU) Finite Frobenius Rings November 7, 2011 17 / 40
Character-Theoretic Proof (c)
I Re-write weight-preservation equation as
n∑j=1
∑a∈R
(aρ)(λ1,j(x)) =n∑
k=1
∑b∈R
(bρ)(λ2,k(x)), x ∈ M .
I Or as
n∑j=1
∑a∈R
ρ(λ1,j(x)a) =n∑
k=1
∑b∈R
ρ(λ2,k(x)b), x ∈ M .
JW (WMU) Finite Frobenius Rings November 7, 2011 18 / 40
Character-Theoretic Proof (d)
I The last equation is an equation of characters on M .
I Characters are linearly independent, so one canmatch up terms (carefully).
I A technical argument involving a preordering givenby divisibility in R shows how to match up termswith units as multipliers.
I This produces a permutation σ and units ui in Rsuch that λ2,k = λ1,σ(k)uk , as desired.
JW (WMU) Finite Frobenius Rings November 7, 2011 19 / 40
Re-Write the Extension Problem
I The character-theoretic proof just given generalizedthe Ward-Wood proof over finite fields.
I Now we will generalize an approach due toMacWilliams; Bogart, Goldberg, and Gordon; andGreferath in order to re-formulate the extensionproblem.
I Will use R-linear codes over an alphabet A, an ideaof Nechaev and his collaborators.
JW (WMU) Finite Frobenius Rings November 7, 2011 20 / 40
Monomial Transformations
I R finite ring, A finite left R-module (an alphabet).
I A linear code over A is a left R-submodule C ⊂ An.
I A monomial transformation T : An → An has theform
T (x1, . . . , xn) = (xσ(1)φ1, . . . , xσ(n)φn),
for (x1, . . . , xn) ∈ An, where σ is a permutation of{1, . . . , n} and φ1, . . . , φn ∈ Aut(A).
JW (WMU) Finite Frobenius Rings November 7, 2011 21 / 40
Re-Formulation of Extension Problem (a)
I View a left R-linear code C ⊂ An as the image of anR-linear homomorphism Λ : M → An, whereΛ = (λ1, . . . , λn) and λi : M → A are R-linear.
I Up to monomial equivalence, what matters is thenumber of λi ’s in a given scale class (under rightaction by automorphisms of A).
I The group Aut(A) of R-automorphisms of A acts onthe right on the group HomR(M ,A) of R-linearhomomorphisms from M to A.
JW (WMU) Finite Frobenius Rings November 7, 2011 22 / 40
Re-Formulation of Extension Problem (b)
I Let O] be the set of nonzero orbits of the action ofAut(A) on HomR(M ,A).
I Let η : O] → N be the multiplicity function thatcounts how many of the λi belong to each scaleclass.
I Functions equivalent to η have appeared elsewhereunder various names (value function, multiset, etc.).
JW (WMU) Finite Frobenius Rings November 7, 2011 23 / 40
Re-Formulation of Extension Problem (c)
I Summary, so far: the monomial equivalence class ofΛ : M → An is encoded by its multiplicity functionη : O] → N.
JW (WMU) Finite Frobenius Rings November 7, 2011 24 / 40
Re-Formulation of Extension Problem (d)
I Now, turn to Hamming weights.
I Note that the Hamming weight depends only on theleft scale class of x ∈ M via units of R :
wt(Λ(ux)) = wt(uΛ(x)) = wt(Λ(x)), x ∈ M , u ∈ U .
I Let O be the set of nonzero orbits of the left actionof the group of units U on M .
JW (WMU) Finite Frobenius Rings November 7, 2011 25 / 40
Re-Formulation of Extension Problem (e)
I The Hamming weight wt(Λ(x)) depends only on thescale classes of the λi (φi ∈ Aut(A)):
wt(Λ(x)) =n∑
i=1
wt(λi(x)) =n∑
i=1
wt(λi(x)φi).
I The Hamming weight does not depend on the orderof the λi .
JW (WMU) Finite Frobenius Rings November 7, 2011 26 / 40
Re-Formulation of Extension Problem (f)
I Let F (O],N) denote the set of all functions fromO] to N. Similarly for F (O,N).
I The Hamming weight gives a well-defined mapW : F (O],N)→ F (O,N):
W (η)(x) =∑λ∈O]
η(λ) wt(λ(x)).
I Summary: the Extension Theorem for Hammingweight holds iff the map W is injective for everyfinite module M .
JW (WMU) Finite Frobenius Rings November 7, 2011 27 / 40
Re-Formulation of Extension Problem (g)
I By formally allowing rational coefficients, we get
W : F (O],Q)→ F (O,Q).
I W is a linear transformation of Q-vector spaces.
I The Extension Theorem for Hamming weight holdsiff the map W is injective for every finite module M .
JW (WMU) Finite Frobenius Rings November 7, 2011 28 / 40
A Counter-Example to Extension (a)
I For R-linear codes defined over a module A, theextension theorem might not hold.
I Let R = Mm(Fq), the ring of m ×m matrices overFq. The group of units is U = GL(m,Fq).
I Let A = Mm,k(Fq), the space of all m × k matrices.A is a left R-module. Aut(A) = GL(k ,Fq).
I Assume m < k .
JW (WMU) Finite Frobenius Rings November 7, 2011 29 / 40
A Counter-Example to Extension (b)
I A general left R-module has the formM = Mm,j(Fq). Then HomR(M ,A) = Mj ,k(Fq) (viaright matrix multiplication).
I Left action of U = GL(m,Fq) on M = Mm,j(Fq):orbits O consist of row reduced echelon matrices ofsize m × j .
I Right action of Aut(A) = GL(k ,Fq) onHomR(M ,A) = Mj ,k(Fq): orbits O] consist ofcolumn reduced echelon matrices of size j × k .
JW (WMU) Finite Frobenius Rings November 7, 2011 30 / 40
A Counter-Example to Extension (c)
I In W : F (O],Q)→ F (O,Q), the dimensions overQ of the domain and range equal the number ofelements in O] and O, respectively.
I dimQ F (O],Q) equals the number of columnreduced echelon matrices of size j × k .
I dimQ F (O,Q) equals the number of row reducedechelon matrices of size m × j .
I Since k > m, dimQ F (O],Q) > dimQ F (O,Q), andW is not injective.
JW (WMU) Finite Frobenius Rings November 7, 2011 31 / 40
Explicit Counter-Examples (a)
I R = M1(Fq) = Fq, A = M1,2(Fq). Remember thatHamming weight depends on elements beingnonzero in A (nonzero as a pair).
I For q = 2, n = 3:
C+ C−(00, 00, 00) (00, 00, 00)(00, 10, 10) (10, 10, 00)(00, 01, 01) (00, 10, 10)(00, 11, 11) (10, 00, 10)
JW (WMU) Finite Frobenius Rings November 7, 2011 32 / 40
Explicit Counter-Examples (b)
I For q = 3, n = 4:
C+ C−(00, 00, 00, 00) (00, 00, 00, 00)(00, 01, 01, 01) (00, 10, 20, 10)(00, 02, 02, 02) (00, 20, 10, 20)(00, 10, 10, 10) (10, 10, 10, 00)(00, 11, 11, 11) (10, 20, 00, 10)(00, 12, 12, 12) (10, 00, 20, 20)(00, 20, 20, 20) (20, 20, 20, 00)(00, 21, 21, 21) (20, 00, 10, 10)(00, 22, 22, 22) (20, 10, 00, 20)
JW (WMU) Finite Frobenius Rings November 7, 2011 33 / 40
Characterizing Finite Frobenius Rings
I Theorem (2008). Suppose R is a finite ring, and setA = R . If the extension theorem for Hammingweight holds for linear codes over R , then R is aFrobenius ring.
I Dinh and Lopez-Permouth (2004–2005) provedsome special cases and developed a strategy toprove the general result.
JW (WMU) Finite Frobenius Rings November 7, 2011 34 / 40
The Strategy of Dinh and Lopez-Permouth
I Every non-Frobenius ring has a copy of someMm,k(Fq) ⊂ Soc(R), with m < k .
I The extension theorem fails for Mm,k(Fq) ⊂ Soc(R),with m < k (as a module over Mm(Fq)).
I View the Mm,k(Fq) counter-examples as modules(and hence counter-examples) over R itself.
JW (WMU) Finite Frobenius Rings November 7, 2011 35 / 40
Structure of a Finite Ring
I Let R be a finite ring with 1.
I R/Rad(R) is a sum of simple rings, which must bematrix rings over finite fields:
R/Rad(R) ∼=⊕
Mmi(Fqi
).
I Soc(RR) is a left module over R/Rad(R), so
Soc(RR) ∼=⊕
Mmi ,ki(Fqi
).
JW (WMU) Finite Frobenius Rings November 7, 2011 36 / 40
Frobenius Rings
I Remember that a finite ring is Frobenius ifR/Rad(R) is isomorphic to Soc(R) as one-sidedmodules (so ki = mi).
I In a non-Frobenius ring, there exist ki 6= mi , withsome larger and some smaller.
I These provide the counter-examples to theextension theorem.
JW (WMU) Finite Frobenius Rings November 7, 2011 37 / 40
Additional Comments (a)
I One can characterize alphabets A for which theextension theorem holds: A ⊂ R plus one morecondition.
I In particular, A = R always satisfies the extensiontheorem for Hamming weight (for any finite ring R ,Frobenius or not). This is a theorem of Greferath,Nechaev, Wisbauer (2004) that extends the originalFrobenius result.
JW (WMU) Finite Frobenius Rings November 7, 2011 38 / 40
Additional Comments (b)
I Some results are known for other weight functions,especially the “homogeneous weight” (again, byGreferath, Nechaev, Wisbauer).
I But, there is much that is not known about otherweight functions. For example, it is not known if theextension theorem is always true for the Lee weightover R = Z/nZ for all n.
I Are there other uses of W : F (O],Q)→ F (O,Q)?
JW (WMU) Finite Frobenius Rings November 7, 2011 39 / 40
References
I These slides and other papers are available on theweb: http : //homepages.wmich.edu/ ∼ jwood
I Many references in the paper “Foundations ofLinear Codes ... ”
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